where \(I_{p\times p}\)  is the \(p\times p\)  identity matrix. Then \(\hat{{\mathbf {\Sigma }}}^{\rm at}_{+}\)  is a positive definite matrix with the same structure as that of \(\hat{{\mathbf {\Sigma }}}^{\rm at}\) . It is easy to show that \(\hat{{\mathbf {\Sigma }}}^{\rm at}_{+}\)  and \(\hat{{\mathbf {\Sigma }}}^{\rm at}\)  attains the same rate of convergence over \(\mathcal {H}(c_{n, p})\) . See Cai, Ren and Zhou [1]} for further discussions.

